Can I build a respectable rig, including periphals, for about $1,000 or less?
Most definately. If you're going that way I, and many others here, would recommend newegg.com as a reliable and reasonably priced online retailer. You can set an account up there and pick and choose the parts for your system to get an total cost. Pretty handy
At the risk of being redundant, i'll give some (AMD specific) advice to get you started

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Motherboard: There are very few manufacturers I would say you should avoid. I personally have had no problems with ECS, Biostar and Gigabyte and have heard nothing but good things about Asus, Abit and MSI. Some main things to consider when choosing a motherboard are:
1. Do you want onboard graphics? With the prices of very good cards dropping all the time, i'd advise buying a less expensive motherboard w/o onboard graphics and spending $80-100 on a decent card.
2. The chipset...VIA's KT333/KT400 the Nforce2 and SIS735 are all good IMO. Others are dated and should be avoided. 3. Make sure it has 3 DDR DIMM slots for easy RAM upgrade options.
4. Ensure it has a 4x or 8x AGP slot, again, upgradability.
5. Onboard LAN is always a plus
6. If it's going to be used to listen to MP3s a lot, then i'd say a seperate sound card would be best, but some say that the Nforce2 has excellent onboard audio.
Graphics: The very acceptable Radeon 8500LE w/128mb ram can be had for $87. I'm a gamer, so I just can't bring myself to recommend a system without a decent graphics card! But who knows, maybe someone will want to game on it sometime.
CPU: Overclocking doesn't sound like it's an option, so the Athlon XP 2400+ (2ghz) should do you proud. With a decent HSF (Glacialtech Igloo, Vantec are two quiet options), shouldn't cost more than $120. Don't forget $7 for the arctic silver III thermal grease. As far as attaching the HSF to the CPU correctly goes, visit
ArcticSilver's website for detailed instructions and you shouldn't go wrong. I'm not sure if it's possible to order a cpu with the HSF already attached other than at a physical computer store.
HDD: I'm sure you know what size would suit your needs, so i'll just recommend Maxtor and Seagate. For standard needs, 2mb cache is just fine.
DVDROM: Yep, DVDROM. No point in buying a standalone CDROM as the DVDROM can read both dvds and cds. Plextor, Lite-On, Teac, Sony, Samsung...the list goes on
CDRW: Probably want one for the mp3s? If so, same brands as listed above should be fine.
Sound card: Of course the Audigy/Audigy2, but don't ignore the excellent price/performance of the Fortissimo 3 for $45.
Speakers: After reading through several different reviews, it seems the most respected manufacturers are Logitech, Klipsch and Altec Lansing. A splendid set of surround sound speakers can be had for $67 (Logitech Z640) all the way up to $170 and way beyond for Klipsch.
Monitor: Whatever size you go for, NEC, Samsung and Viewsonic are good choices.
RAM: 512 of Crucial PC2700 can be had for $75 from their own website. However, if you go for an Nforce2 board, research ram more thoroughly as i've heard of issues between Crucial's ram and that chipset (specifically the top end Asus board).
Printers: HP and Canon are probably the best in the biz.
Good luck